Optimal Allocation of Observations in Inverse Linear Regression

Abstract

Consider the problem of estimating $x$ under the inverse linear regression model $Y_i = \alpha + \beta x_i + \varepsilon_i,\quad Z_j = \alpha + \beta x + \varepsilon_j'$ for $i = 1,\cdots, n,\cdots, j = 1,\cdots, m,\cdots,$ where $\{\varepsilon_i\}, \{\varepsilon_j'\}$ are two sequences of i.i.d. rv's with 0 means and finite variances, $\{x_i\}$ is a sequence of known constants and $\alpha, \beta, x$ are unknown parameters. For fixed $T = m + n$, this paper considers a sequential procedure for the optimal allocation of $m$ and $n$. It is shown that, as $T \rightarrow \infty$, the procedure is asymptotically optimal.